We point out that the entanglement between the emitters and the waveguide mode can be markedly incommensurate for initial states \(|\psi _{3}(0)\rangle \)  and \(|\psi _{4}(0)\rangle \)  in the case of \(\theta =0\) . This can be verified by calculating the linear entropy \(S=1-\textrm {Tr}(\rho ^{2})\) , which estimates here the correlation between the emitters and the electromagnetic field in the waveguide [1]}, [2]}. \(\rho \)  denotes the reduced density matrix of the emitters, which can be obtained by taking a trace over the waveguide states, i.e., \(\rho =\textrm {Tr}_{\textrm {w}}[|\psi (t)\rangle \langle \psi (t)|]\)  with \(|\psi (t)\rangle \)  given in Eq. (REF ). We plot in Figs. REF (e) and REF (f) the linear entropies \(S\)  (corresponding to initial state \(|\psi _{3}(0)\rangle \) ) and \(S^{\prime }\)  (corresponding to initial state \(|\psi _{4}(0)\rangle \) ) for \(\theta =\pi /2\)  and \(\theta =0\) , respectively. It can be found that for \(\theta =0\)  the evolutions of linear entropy can be quite different by initially exciting different emitters (\(a\)  or \(b\) ) to the excited state while for \(\theta =\pi /2\)  there is no significant difference between the two cases. We also plot in Figs. REF (e) and REF (f) the total populations of all emitters (i.e., survival probabilities) \(P_{\textrm {tot}}=P_{a,3}+P_{b,3}+P_{c,3}\)  and \(P_{\textrm {tot}}^{\prime }=P_{a,4}+P_{b,4}+P_{c,4}\)  for \(\theta =\pi /2\)  and \(\theta =0\) , respectively. As expected, the total populations exhibit similar behaviors with those of the linear entropies [1]}. Such a result is reminiscent of the nonreciprocal entanglement demonstrated in Ref. [4]}, which reveals that the conditions of nonreciprocal entanglement and transport are not necessary the same. However, we refer to the phenomenon here as incommensurate entanglement to distinguish it from the phenomena stemming from broken time-reversal symmetry. As a final note, the incommensurate entanglement vanishes in the Markovian limit, where for \(\theta =m\pi \)  the excitation is transferred equally from the initial emitter to the other two and comes back with the same probability, independent of the initial state.
